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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 130130h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 130130.h3 | 130130h1 | \([1, 0, 1, -9468, 6899258]\) | \(-19443408769/4249907200\) | \(-20513490322124800\) | \([2]\) | \(1244160\) | \(1.8093\) | \(\Gamma_0(N)\)-optimal |
| 130130.h2 | 130130h2 | \([1, 0, 1, -604348, 179176506]\) | \(5057359576472449/51765560000\) | \(249862470898040000\) | \([2]\) | \(2488320\) | \(2.1558\) | |
| 130130.h4 | 130130h3 | \([1, 0, 1, 85172, -185863494]\) | \(14156681599871/3100231750000\) | \(-14964226512985750000\) | \([2]\) | \(3732480\) | \(2.3586\) | |
| 130130.h1 | 130130h4 | \([1, 0, 1, -4413608, -3468173382]\) | \(1969902499564819009/63690429687500\) | \(307421539229492187500\) | \([2]\) | \(7464960\) | \(2.7051\) |
Rank
sage: E.rank()
The elliptic curves in class 130130h have rank \(1\).
Complex multiplication
The elliptic curves in class 130130h do not have complex multiplication.Modular form 130130.2.a.h
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
